Question

Foe three non zero vectors a,b and c , prove that [a-b.b-c.c-a]=0

Answer 1

Answer:

0

Step-by-step explanation:

We need only two facts to know about the box product, a.k.a scalar triple product

1. [a,b,c] is given by the determinant of a matrix, whose first, second and third rows are the components of the vectors, a, b and c respectively.

2. This means, all properties of determinants are applicable for box product. In particular the row transforms.

Now, let us use the row transformation, R1 -> R1 + R2 + R3. This make the first row of the determinant to be = (a-b) + (b-c) + (c-a) = 0

Now, the given box product becomes the determinant of a matrix whose first row is all zeros, which makes the value of the determinant zero. Hence the proof!